In the discrete setting, amenable hyperbolic groups are virtually cyclic, but in the locally compact setting, it is a much wider and more interesting class. In a joint work with Caprace and Monod, we give a structural characterization of amenable hyperbolic locally compact groups, in terms of contracting automorphisms of locally compact groups. If time permits, we\'ll indicate how L^p-cohomology provides some partial results about the QI classification of these groups. The talk will start at an elementary level.